Minimum diameter of set inscribed in a unit sphere For a study of the stability of certain maps taking values in a sphere I have the following question.
Let $A$ be a subset of $\mathbb{R}^n$. Suppose $A$ lies in a unit ball, but in no ball of smaller radius centered at any point. I'm interested in bounding the diameter of $A$ below. I conjecture that a regular $n$-dimensional simplex (inscribed in the unit sphere) minimizes this diameter. It's not hard to show that the diameter of such a simplex, when indexed by $n$, forms a strictly decreasing sequence converging to $\sqrt{2}$. Is $\sqrt{2}$ a strict lower bound for general $A$?
I wouldn't be surprised if this problem were solved long ago. I would like a reference I could cite in a paper.
 A: It is Jung's theorem. If the diameter would be less than
$d=\sqrt{\frac{2(n+1)}{n}}$, there would be a smaller enclosing ball.
https://en.wikipedia.org/wiki/Jung%27s_theorem
Supplement: In the plane, there is a remarkable strengthening of Jung:
Every figure of unit diameter can be enclosed by a regular hexagon such that its opposite sides are at a distance of $1/\sqrt3$.
http://gymarkiv.sdu.dk/MFM/kdvs/mfm%201-9/mfm-3-2.pdf.
This usually is the main step in the solution of Borsuk's problem in the plane.
https://en.wikipedia.org/wiki/Borsuk%27s_conjecture
A: $\sqrt{2}$ is indeed a strict lower bound. A fairly short proof goes as follows:
Working with closed sets, we can assume $A$ to be a
compact set of the closed unit ball $B$ centered at $0$,
perhaps after a suitable translation. No closed ball of
radius strictly smaller than $1$ and arbitrary center
contains $A$.
By compacity of $A$, it intersects $B$ in a point
which we can assume to be the first coordinate $e_1$
of an orthonormal basis.
Suppose now that $A$ has exactly diameter $\sqrt{2}$.
Suppose first that $A$ contains no unit vector orthogonal to $e_1$. By compacity of $A$, we can
therefore assume that there exists $\epsilon>0$ such
that every element of the intersection of $A$ with the unit sphere $S$ has scalar product $\geq \epsilon$ with $e_1$. This implies that the closed halfsphere
defined by all unit-vectors making an obtuse
angle with $e_1$ is at strictly positive distance from $A$. We can therefore move the center of the unit ball
by a small amount in the direction of $e_1$ in order
to get a unit ball containing $A$ in its interior.
We can therefore assume that $A$ contains a second
element $e_2$ of an orthonormal basis. The same argument
as above shows that we can find a smaller sphere
if the $n-1$ dimensional halfsphere of unit vectors making obtuse angles with $e_1+e_2$ does not intersect $A$:
Move the center of $U$ by a tiny amount in direction $e_1+e_2$. We can therefore assume the existence
of a unit vector $e_3$ in $A$ making an obtuse angle with $e_1+e_2$. Since $A$ has diameter $\sqrt 2$, the
element $e_3$ is orthogonal to $e_1$ and $e_2$.
Iteration of
this argument shows that $A$ contains an orthonormal basis $e_1,\ldots,e_n$. But now we can apply our
favourite argument once more and move the
center of the unit ball by a tiny amount in the direction of $e_1+\ldots+e_n$ such that the moved unit
ball contains $A$ in its interior.
A: Your conjecture that a regular simplex minimizes the diameter is correct.
Let $M$ be the intersection of $A$ with the boundary of the minimum ball containing $A$. Then the convex hull of $M$ contains the center of the ball. By Caratheodory's theorem there are at most $n+1$ points in $M$ whose convex hull contains the center. It remains to prove that any inscribed simplex that contains the center in its interior has at least one edge longer than the edge of a regular inscribed simplex. (If less than $n+1$ points from $M$ were taken, then the dimension reduces; alternatively, just add copies of points.)
This should have been done somewhere. Here is a sketch of a proof. If all edges incident to a vertex of an inscribed simplex (containing the sphere center) are shorter than the edge of a regular inscribed simplex, then the opposite face is closer to the center than a face of a regular inscribed simplex. Then apply induction on the dimension.
Edit: for a more general statement on edge lengths of inscribed simplices see Ilya Bogdanov's answer to this question.
